News on B K µ + µ Danny van Dyk B Workshop Neckarzimmern February 19th, 2010 Danny van Dyk (TU Dortmund) News on B K µ + µ 1 / 35
The Goal 15 1 10 0.8 5 0.6 C10 0-5 0.4-10 0.2-15 -15-10 -5 0 5 10 15 C 9 0 Danny van Dyk (TU Dortmund) News on B K µ + µ 2 / 35
Outline From Scratch to a Partonic Matrix Element From Quarks to Mesons From Matrix Elements to Observables News from Fits at Large Dimuon Invariant Mass Danny van Dyk (TU Dortmund) News on B K µ + µ 3 / 35
From Scratch to a Partonic Matrix Element Outline From Scratch to a Partonic Matrix Element From Quarks to Mesons From Matrix Elements to Observables News from Fits at Large Dimuon Invariant Mass Danny van Dyk (TU Dortmund) News on B K µ + µ 4 / 35
From Scratch to a Partonic Matrix Element Motivation Why is B K µ + µ so damn interesting? Transition at quark level: b sµ + µ Flavor Changing Neutral Current (FCNC), forbidden at tree level in the Standard Model (SM) Needs loops, which allow heavy particles to enter virtually These heavy particles might stem from Beyond the SM (BSM) Danny van Dyk (TU Dortmund) News on B K µ + µ 5 / 35
From Scratch to a Partonic Matrix Element Effective Theory H eff (b sµ + µ, b sγ) = 4G F 2 V tb V ts Similar to 4-Fermi interaction C i (µ)o i (µ) + O (V ub Vus) Calculate the complete theory (SM, SUSY) at a large energy scale (M W ) Match coefficient C i (M W ) onto theory Calculate C i (m b ) from C i (M W ) via QCD running. i Danny van Dyk (TU Dortmund) News on B K µ + µ 6 / 35
From Scratch to a Partonic Matrix Element Operator Product Expansion C i (µ)o i (µ) i Operators O i have mass dimension 6 not renormalizable Coefficients C i are called Wilson coefficients C i are running couplings of new effective interations C i are real numbers in this work (and in models with Minimal Flavor Violation) Danny van Dyk (TU Dortmund) News on B K µ + µ 7 / 35
From Scratch to a Partonic Matrix Element Operators e.g. [ sγ i µq] [ qγ i,µ b] s q e (4π) 2 m b [ sσ µν P R b] F µν s O 1...6 O 7 γ b q b s µ s µ O 9 O 10 b µ b µ e 2 [ sγ (4π) 2 µ P L b] [ µγ µ e µ] 2 [ sγ (4π) 2 µ P L b] [ µγ µ γ 5 µ] Danny van Dyk (TU Dortmund) News on B K µ + µ 8 / 35
From Scratch to a Partonic Matrix Element Wilson Coefficients in the SM Calculated from α s, m t, M W and θ W At µ = m b 4.8 GeV C 1-0.257 C 6 +0.001 C 2 +1.009 C 7-0.298 C 3-0.005 C 8-0.164 C 4-0.078 C 9 +4.211 C 5 0 C 10-4.103 Danny van Dyk (TU Dortmund) News on B K µ + µ 9 / 35
From Scratch to a Partonic Matrix Element Our First Matrix Element For b sµ + µ s µ s µ s µ b O 9 µ b O 10 µ b O 7 γ µ im = G F V tb V α { e ts 2 π C 9 [ sγ µ P L b ] [ µγ µ µ] + C 10 [ sγ µ P L b ] [ µγ µ γ 5 µ] 2m b q ν C 7 [ sσ µν q 2 P R b ] } [ µγ µ µ] Danny van Dyk (TU Dortmund) News on B K µ + µ 10 / 35
From Quarks to Mesons Outline From Scratch to a Partonic Matrix Element From Quarks to Mesons From Matrix Elements to Observables News from Fits at Large Dimuon Invariant Mass Danny van Dyk (TU Dortmund) News on B K µ + µ 11 / 35
From Quarks to Mesons Naive Factorization Factorize leptonic and hadronic parts µ + µ K H eff B µ + Γ µ K sγb B Take the hadronic matrix element: [ sγb] K (k, η) sγb B(p) Factor out the Lorentz structure, e.g. K (k, η) sγ µ b B(p) 2V (q2 ) m B + m V ε µρστ η,ρ p σ k τ, q p k Form factor V (q 2 ) covers all hadronic contributions Danny van Dyk (TU Dortmund) News on B K µ + µ 12 / 35
From Quarks to Mesons Hadronic Form Factors Total of 7 form factors (reduceable by symmetries) in B K µ + µ : K (k, η) sγ µ b B(p) = 2V (q2 ) m B + m V ε µρστ η ρ p σ k τ K (k, η) sγ µ γ 5 b B(p) = iη ρ[ 2m V A 0 (q 2 ) q ( µq ρ q 2 + (m B + m V )A 1 (q 2 ) g µρ q ) µq ρ q 2 A 2 (q 2 q ( ρ ) (p + k) µ m2 B )] m2 V m B + m V q 2 (p k) µ K (k, η) siσ µν q ν b B(p) = 2T 1 (q 2 )ε µρστ η ρ p σ k τ ) K (k, η) siσ µν γ 5 q ν b B(p) = it 2 (q 2 ) (η µ(m B 2 mv 2 ) (η q)(p + k) µ ( )( + it 3 (q 2 ) η q 2 ) q q µ mb 2 (p + k) µ m2 V Danny van Dyk (TU Dortmund) News on B K µ + µ 13 / 35
From Quarks to Mesons OPE in 1/ q 2 So far: Operator Product Expansion (OPE) in 1/M 2 W G F Now additionally: OPE in 1/m b and 1/ q 2 q 2 = dilepton invariant mass, 4m 2 µ q 2 (M B M K ) 2 Uses Heavy Quark Effective Theory (HQET) Result: Systematic approach, model independent up to Λ/m b, Λ/ q 2, m 2 c/m 2 b, m2 c/q 2 (B. Grinstein, D. Pirjol 04) Valid for large q 2 O(m 2 b ) Danny van Dyk (TU Dortmund) News on B K µ + µ 14 / 35
From Quarks to Mesons Meson Matrix Elements im(b K µ + µ ) = G F V tb V α e ts 2 π { C9 eff K (k, η) sγ µ P L b B(p) [ µγ µ µ] + C 10 K (k, η) sγ µ P L b B(p) [ µγ µ γ 5 µ] C7 eff 2m b q } ν q 2 K (k, η) sσ µν P R b B(p) [ µγ µ µ] Danny van Dyk (TU Dortmund) News on B K µ + µ 15 / 35
From Matrix Elements to Observables Outline From Scratch to a Partonic Matrix Element From Quarks to Mesons From Matrix Elements to Observables News from Fits at Large Dimuon Invariant Mass Danny van Dyk (TU Dortmund) News on B K µ + µ 16 / 35
From Matrix Elements to Observables Kinematics B K ( Kπ) q( µ + µ ) Danny van Dyk (TU Dortmund) News on B K µ + µ 17 / 35
From Matrix Elements to Observables To B K ( Kπ) µ + µ Depends on 4 kinematic variables: q 2, θ l, θ K, φ Differential decay width is 4-differential: d 4 Γ dq 2 d cos θ l d cos θ K dφ = 3 8π J(q2, θ l, θ K, φ) J(q 2, θ l, θ K, φ) can be expanded in the angles J(q 2, θ l, θ K, φ) = J i (q 2 )f i (θ l, θ K, φ) Total of 9 angular coefficients: J 1,..., J 9 J i can be expressed through transversity amplitudes A 0,, Danny van Dyk (TU Dortmund) News on B K µ + µ 18 / 35
From Matrix Elements to Observables Transversity Amplitudes What are A 0,,? 1. Project out helicity amplitudes M = η µ K (λ)l ν M µν L η q H λ η µ K (λ)η ν q (λ)m µν L ν : Lepton current 2. Build linear combinations of helicity amplitudes H 0,± A, = (H + H )/ 2 A 0 = H 0 (Signs may change depending upon convention) Danny van Dyk (TU Dortmund) News on B K µ + µ 19 / 35
From Matrix Elements to Observables Differential Decay Width / Longitudinal Fraction Differential Decay Width dγ/dq 2 : (For massless leptons) dγ/dq 2 = J 1 J 2 3 = A 2 + A 2 + A 0 2 Longitudinal Fraction F L : Longitudinal K versus total K F L = A 0 2 dγ/dq 2 Danny van Dyk (TU Dortmund) News on B K µ + µ 20 / 35
From Matrix Elements to Observables Forward-Backward Asymmetry Definition ( 1 +1 d 2 Γ A FB = dγ/dq 2 d cos θ l 0 dq 2 d cos θ l Can be written as 1 A FB = J 6 dγ/dq 2, J 6 Re { A A } A FB (C 9 + κ m2 b q 2 C 7 ) C 10 (C 9 + κ m2 b q 2 C 7 ) 2 + C 2 10 0 d 2 ) Γ d cos θ l dq 2 d cos θ l At q 2 m 2 b : Tests if C 9 C 10 is SM-like Danny van Dyk (TU Dortmund) News on B K µ + µ 21 / 35
From Matrix Elements to Observables Data (as of February 2010) Measured are: dγ/dq 2, A FB and F L Measured by: BaBar, Belle and CDF Few to very few bins with few events per bin Theorists want: Measurement of all J i Danny van Dyk (TU Dortmund) News on B K µ + µ 22 / 35
News from Fits at Large Dimuon Invariant Mass Outline From Scratch to a Partonic Matrix Element From Quarks to Mesons From Matrix Elements to Observables News from Fits at Large Dimuon Invariant Mass Danny van Dyk (TU Dortmund) News on B K µ + µ 23 / 35
News from Fits at Large Dimuon Invariant Mass db/dq 2 (B K l + l ), l = e, µ Preliminary Result for db/dq 2 at Low Recoil, BaBar 06, Belle 09, CDF 09 db/dq 2 [10 7 ] 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Extrapolation J/ψ ψ 0 2 4 6 8 10 12 14 16 18 q 2 [GeV 2 ] Danny van Dyk (TU Dortmund) News on B K µ + µ 24 / 35
News from Fits at Large Dimuon Invariant Mass A FB (B K l + l ), l = e, µ Preliminary Result for A FB at Low Recoil, BaBar 08, Belle 09, CDF 09 AFB 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 C 9 C10 is SM-like Extrapolation J/ψ ψ 0 2 4 6 8 10 12 14 16 18 q 2 [GeV 2 ] Danny van Dyk (TU Dortmund) News on B K µ + µ 25 / 35
News from Fits at Large Dimuon Invariant Mass F L (B K l + l ), l = e, µ Preliminary Result for F L at Low Recoil, BaBar 08, Belle 09, CDF 09 FL 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 Extrapolation J/ψ ψ 0 2 4 6 8 10 12 14 16 18 q 2 [GeV 2 ] Danny van Dyk (TU Dortmund) News on B K µ + µ 26 / 35
News from Fits at Large Dimuon Invariant Mass The Result Preliminary Result: Scan for C 9,10, plotted is L normalized to max. 15 1 15 1 10 0.8 10 0.8 5 0.6 5 0.6 C10 0 C10 0-5 0.4-5 0.4-10 0.2-10 0.2-15 -15-10 -5 0 5 10 15 0-15 -15-10 -5 0 5 10 15 0 From db/dq 2, Belle+CDF C9 C9 From A FB, Belle+CDF : SM values Danny van Dyk (TU Dortmund) News on B K µ + µ 27 / 35
News from Fits at Large Dimuon Invariant Mass The Result Preliminary Result: Scan for C 9,10, plotted is L normalized to max. 15 1 10 0.8 5 0.6 C10 0-5 -10 0.4 0.2-15 -15-10 -5 0 5 10 15 0 C 9 Danny van Dyk (TU Dortmund) News on B K µ + µ 28 / 35
News from Fits at Large Dimuon Invariant Mass Conclusion Large q 2 can be explored We calculated J i, A 0,, for large q 2 sign (C 9 C10 ) could be determined; is SM-like More data and orthogonal data needed... Enhanced precisions needed...... to test the SM... to see where it fails Danny van Dyk (TU Dortmund) News on B K µ + µ 29 / 35
Literature Literature on this decay includes (amongst others) Overview (W. Altmannshofer et al) arxiv:0811.1214 [hep-ph] Overview (A. Ali et al) arxiv:hep-ph/9910221 CP Asymmetries (C. Bobeth et al) arxiv:0805.2525 [hep-ph] 1/ q 2 Expansion (B. Grinstein, D. Pirjol) arxiv:hep-ph/0404250 Danny van Dyk (TU Dortmund) News on B K µ + µ 30 / 35
Backup Slides Outline Backup Slides Danny van Dyk (TU Dortmund) News on B K µ + µ 31 / 35
Backup Slides Why no O 8? s µ O 8 g? b µ µ µ is colorless g carries color! Danny van Dyk (TU Dortmund) News on B K µ + µ 32 / 35
Backup Slides QCD Factorization (QCDF) + Soft Collinear Effective Theory (SCET) Replace C 7 T i (q 2 ) T i (q 2 ), C eff 9 (q2 ) C 9 All q q-loop contributions can now be resorbed into T i (q 2 ) Applicable at q 2 6 7 GeV 2 At leading order only 2 form factors ξ, Danny van Dyk (TU Dortmund) News on B K µ + µ 33 / 35
Backup Slides Form Factor Values Small q 2 (Large Recoil) 2 independent form factors in QCD Factorization (QCDF) + Soft Collinear Effective Theory(SCET) Obtained from Light Cone Sum Rules (LCSR) and consequent fits Uncertainties of 10 20% (theory and input) In agreement with measurements for q 2 = 0 Large q 2 (Low Recoil) 4 independent form factors at LO Obtained from Lattice QCD calculations Danny van Dyk (TU Dortmund) News on B K µ + µ 34 / 35
Backup Slides Non-factorizable contributions Operators O 1...6 are not factorizable Example [ sγ µ b][ qγ µ q] O 7,9,10 are completely factorizable! In B K ( ) µ + µ most non-factorizable contributions only appear at NLO in α s. Resorb non-factorizable contributions into effective Wilson coefficients C eff 7,9 Danny van Dyk (TU Dortmund) News on B K µ + µ 35 / 35